Facility Location
July 17, 2022 | Author: Anonymous | Category: N/A
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Facility Location •
Logistics Management
•
Factors that Affect Location Decisions
• •
Distance Measures Classification of Planar Facility Location Problems
•
Planar Single-Facility Location Problems – Minisum Location Problem with Rectilinear Distances – Minisum Location Problem with Euclidean Distances – Minimax Location Problem with Rectilinear Distances – Minimax Location Problem with Euclidean Distances
•
Planar Multi-Facility Location Problems
– Minisum Location Problem with Rectilinear Distances
Logistics Management •
Logistics Management can be defined as the managem management ent of the transportation and distribution of goods. The term goods includes raw mater materials ials or subassemblies obtained from suppliers as well as finished goods shipped ship ped from plants to warehouses or customers.
•
Logistics management problems can be classified into three categories – Location Problem Problems s in!ol!e determining thecost location of one each or more facilities in one or more of se!eral potential sites. The of locating newnew facilit" at each of the potential sites is assumed to be #nown. $t is the fixed cost of locating a new facilit" at a particular site plus the operating and transportation cost of ser!ing customers from this facilit"%site combination.
– Allocation Problem Problemss assume that the number and location of facilities are #nown a priori and attempt attempt to determ determine ine how each custom customer er is to be ser!e ser!ed. d. $n other words& gi!en the demand for goods at each customer center& center& the production or suppl" capacities at each facilit"& and the cost of ser!ing each customer from each facilit"& the allocation problem determines how much each facilit" is to suppl" to each customer center. – Location-Allocation Problem Problemssin!ol!e determining not onl" how much each customer is to recei!e from each facilit" but also the number of facilities along with their locations and capacities.
Factors that Affect Location Decisions •
Proximit" to source of raw materials.
•
'ost and a!ailabilit" of energ" and utilities.
•
'ost& a!ailabilit"& s#ill& and producti!it" of labor.
•
(o!ernment regulations at the federal& state& count"& and local le!els.
•
Taxes Ta xes at the federal& state& count"& and local le!els.
•
$nsurance.
• •
'onstruction costs and land price.
•
Exchange rate fluctuation.
•
Export and import regulations& duties& and tariffs.
•
Tr Transportation ansportation s"s s"stem. tem.
• •
Te Technical chnical expertise. En!ironmental regulations at the federal& state& count" and local le!els.
•
)upport ser!ices.
•
'ommunit" ser!ices % schools& hospitals& recreation& and so on.
•
*eather.
•
Proximit" to customers.
•
+usiness climate.
•
'ompetition%related factors.
(o!ernment and political stabilit".
Distance Measures •
Rectilinear distance ,L- norm
Pi 0 ,ai& bi
– d,/& Pi 0 1x % ai1 2 1" % bi1 / 0 ,x& "
•
)traight line or Euclidean distance ,L 3 norm – d,/& Pi 0
,x % a i
3
2 ," % b i
Pi 0 ,ai& bi
3
/ 0 ,x& "
•
Tcheb"she! distance ,L∞ norm
Pi 0 ,ai& bi
– d,/& Pi 0 max41x % ai1& 1" % bi15
/ 0 ,x& "
Classification of Planar Facility Location Problems 7 of facilities
8b9ecti!es
Distance measures
Rectilinear Minisum
Euclidean Tcheb"she!
)ingle% 6acilit"
Rectilinear Minimax
Euclidean Tcheb"she!
6acilit" Location
Rectilinear Minisum
Euclidean Tcheb"she!
Multi% 6acilit"
Rectilinear Minimax
Euclidean Tcheb"she!
Planar Single-Facility Location •
Minisum Formulation Min f,x 0
Formulations
m
∑ w i × d( /& Pi ) i = - location of the new facilit" where / 0 ,x& "
Pi 0 ,ai& bi location of the i%th existing facilit"& i 0 -& :& m wi weight associated to the i%th existing facilit" 6or example& wi 0
c
t i& i
i where ci cost per hour! o f tra!el&
ti number of trips per month&
!i a!erage !elocit". •
Minimax Formulation Min f,x 0 Max
s. t.
4w i × d,/& Pi5
wi × d,/&i 0P-&i:&≤m ;& i 0 -& :& m
⇔ Min
;
Insights for the Minisum Problem Problem with Eucliean Distance ?ole P3
P?ori;ontal pegboard
P>
P< P=
w<
w-
w3
w> w=
•
)tring
*eight proportional to wi
Ma!ority "heorem "heorem *hen one weight constitutes a ma9orit" of the total weight& an optimal new facilit" location coincides with the existing facilit" which has the ma9orit" weight.
Minisum Location Problem with #ectilinear Distances Min f,x& " 0
m
∑
i0 -
@ote that
w
i
×
A1x
f -,x 0
∑
w
i
1x
−
a i1
w
i
1"
−
b i1
i0 m
f 3," 0
a i 1+ 1 "
−
b i 1B
f,x& " 0 f -,x 2 f 3," m
where
−
∑
i0 -
The cost of mo!ement in the x direction is independent of the cost of mo!ement in the " direction& and !ice!ersa. @ow& we loo# at the x direction. f -,x is con!ex ⇔ a local min is a global min.
Minisum Location Problem with #ectilinear Distances $cont%& •
The coordinates of the existing facilities are sorted so that a- ≤ a3 ≤ a> ≤ :.
• @ow& we consider the case of m 0 >. 'ase x ≤ a- f -,x 0 w- 1a- % x1 2 w 3 1a3 % x1 2 w > 1a> % x1 0 % ,w- 2 w3 2 w>x 2 w- a- 2 w3 a3 2 w> a> 0 % * x 2 w - a- 2 w3 a3 2 w> a>& where * 0 w- 2 w3 2 w> 'ase a- ≤ x ≤ a3 f -,x 0 w- 1a- % x1 2 w 3 1a3 % x1 2 w > 1a> % x1 0 ,w- % w3 % w>x % w- a- 2 w3 a3 2 w> a> 0 ,% * 2 3 w - x % w- a- 2 w3 a3 2 w> a> :
'b!ecti(e Function f )$x&
f -,x
% w % w % w -
3
>
The slope changes sign w- 2 w3 2 w>
w- % w3 % w> w- 2 w3 % w>
ww3 w> a-
a3
a>
x
Minisum Location Problem with #ectilinear Distances $cont%& •
Slo*es of f )$x& +
MC 0 % ,w- 2 w3 2 w> 0 % * M- 0 3 w - 2 MC M3 0 3 w 3 2 MM> 0 3 w > 2 M3 0 w - 2 w3 2 w> 0 * •
Meian conitions +
f -,x is minimi;ed at the point where the slope changes from nonpositi!e to nonnegati!e. M- 0 w- % w3 % w> C ⇔ 3 w- ,w- 2 w3 2 w> 0 * w- *3 M3 0 w- 2 w3 % w> ≥ C ⇔ 3 ,w- 2 w3 ≥ ,w- 2 w3 2 w> 0 * ,w- 2 w3 ≥ *3
Exam*le ) • Problem Data + m0> a- 0 -C
a3 0 3C a> 0 =C
w- 0 <
w3 0 F
w> 0 =
• Solution + * 0 w- 2 w3 2 w> 0 -< *3 0 G.< w- 0 < G.< w- 2 w3 0 -- H G.< ⇒
Minimi;ing point a3 0 3C
Linear Programming Formulation Min f -,x 0 w- 1a- % x1 2 w 3 1a3 % x1 2 w > 1a> % x1
⇔
Min ; 0 w- ,r -2 s- 2 w3 ,r 32 s3 2 w> ,r >2 s>&
Dual !ariables
s. t. x % r -2 s-
% r 32 s3 0 a3&
x
% r >2 s> 0 a>&
0 a-&
" - x
" 3
" >
r 9& s 9 ≥ C& 9 0 -& 3& >. a 9 % x 0 r 9 % s 9 & 1a 9 % x1 0 r 9 2 s 9& r 9& s 9 ≥
•
Relationships among !ariables C.
•
$f both r 9& s 9 H C& we can reduce each b" ε 9 0 min 4r 9& s 95.
•
This maintains feasibilit" and reduces ; ⇒ $n an optimal solution& at least one of the r 99 and s 9 is C& i. e.& r 9 × s 9 0 C.
Linear Programming Programming Formulation $cont%& •
Dual Problem Max g 0 % a-"- % a3 "3 % a> "> 2 ,w- a- 2 w3 a3 2 w> a>
s. t. "- 2 "3 2 "> 0 w- 2 w3 2 w> 0 * C ≤ " 9 ≤ 3 w 9& 9 0 -& 3& > ⇔ Min a-"- 2 a3 "3 2 a> "> s. t. "- 2 "3 2 "> 0 *
C ≤ " 9 ≤ 3 w 9& 9 0 -& 3& > C ≤ "- ≤ 3 w
C ≤ "3 ≤ 3 w3
* -
•
'omplementar" slac#ness conditions
a-
a3
a>
C ≤ "> ≤ 3 w>
C " 9I 3 w 9 ⇒ xI 0 a 9
a- ≤ a3 ≤ a>
* 3
Exam*le ) + Dual Solution •
f -,x 0 I 0 C C " 3I -3 ⇒
xI 0 a3 0 3C
-< 3
Minisum Location Problem with Eucliean Distances m
•
Min f,x& " 0
∑
i0 -
w
i
×
A,x
−
a i
3
+
,"
−
3
b i B
3
,ai& bi
The optimum location is alwa"s in the con!ex hull of 4,a-& b-& :& ,am& bm5
•
'olinear case all the points are in a line
,ai& bi
⇒ The problem reduces to minimi;ing f -,x& which is the rectilinear distance
problem.
,on-colinear Case -
•
The graph of A , x
−
a i
3
+
,"
−
3
b i B 3 is a cone ,strictl" con!ex function.
3 3
3
"
A , x − a i . + , " − b i . B
,ai& bi
contours
x ,ai& bi& C
"
-
m
∑ •
x
w
i
×
A,x
−
a i
3
+
,"
−
f,x& " 0 i 0 con!ex hull is a line segment.
3
b i B3
is strictl" con!ex unless the
,on-colinear Case $cont%& •
6irst order optimalit" conditions ∂ f , x & " ∂ x
= x
C
⇒
∂ f , x & " ∂ "
•
C
= "
,xC& "C is optimal
C
C
Kn" point where the partial deri!ati!es are ;ero is optimal. Let
γ i , x & "
w
=
-
A,x
−
a i
3
m
and
Γ ,x &
"
i
= ∑ - γ i , x & i
=
"
+
,"
−
3
b i B3
,on-colinear Case $cont%& m ∂ f , x & " = ∑ γ i , x & " i0 ∂ x
∂ f , x & "
m
= ∑ γ i , x & "
∂ "
,x % a i 0C
×
," % b i 0C
i0 -
m
⇒
×
x
=
∑
a
i0 -
i
=
γ i , x & "
Γ ,x&" m
"
×
∑
i0 -
b
i
×
γ i , x & "
Γ ,x&"
,o-colinear Case $cont%& γ i , x & " . =
∞
•
$f the optimal solution is in an exiting facilit" ,a i& bi& then
•
K simple wa" to a!oid the problem of di!ision b" ;ero is to perturb the problem as follows -
m
M in f,x & "
= ∑
i -
w
i
×
A,x
−
a i3
+
,"
−
b i3
+
δ B
3
=
where δ H C and small.
f,x&"
f,x&" is flat near the optimum.
x
,xI&"I "
eis.fel e is.fel/s /s Algorithm $nitiali;ation , i , x & " C
C
C
C
or , i i , x & "
-
m
m
i -
= ∑ =
,a i& b i
=
*
m
∑
w
i
i -
×
,a i& b i&
w h e re
m
$terati!e step ,# 0 -& 3& :
∑
x
#
0
i0 -
a i γ i , x
∑ "
#
0
b i γ i , x
i0 -
Γ
,i ,x # & " # % ,x
or , i i f , x
# %-
# %-
&"
# %-
# %-
Γ , x # %- & " # %- m
Terminating conditions
*
=
&"
# %-
,x # %-
# %-
&"
# %-
&" # %-
&" # %-
% f,x # & " #
≤
ε
≤
ε
0 w
-
2 ... 2 w
m
Exam*le 0 Problem Data +
m0= P- 0 ,C& C
w- 0 -
P> 0 , 0 -
P3 0 ,C& -C P= 0 ,-3& F
w3 0 -
w= 0 -
Solution + xC 0 ,
,3
,-
,=
1x % ai1 2 1" % bi1 ≤ ; % hi ⇔ x % ai 2 " % bi ≤ ; % hi
,-
ai % x 2 bi % " ≤ ; % hi
,3
ai % x 2 " % bi ≤ ; % hi x % ai 2 bi % " ≤ ; % hi
,> ,=
Minimax Location Problem with #ectilinear Distances $cont%& Min ; s. t .
x 2 " % ; ≤ ai 2 bi % hi&
i 0 -& ...& m
x 2 " 2 ; ≥ ai 2 bi 2 hi&
i 0 -& ...& m
% x 2 " % ; ≤ % ai 2 bi % hi& % x 2 " 2 ; ≥ % ai 2 bi 2 hi&
i 0 -& ...& m i 0 -& ...& m
⇔
Min ; s. t .
x 2 " % ; ≤ c- wh where
c- 0 Min 4ai 2 bi % hi5 i 0 -& ...& m
x 2 " 2 ; ≥ c3
c3 0 Max 4ai 2 bi 2 hi5
% x 2 " % ; ≤ c>
c> 0 Min 4% ai 2 bi % hi5
% x 2 " 2 ; ≥ c=
c= 0 Max 4% ai 2 bi 2 hi5i 0 -& ...& m
i 0 -& ...& m
i 0 -& ...& m
Minimax Location Problem with #ectilinear Distances $cont%& Min ; s. t. % x % " 2 ; ≥ % c x 2 " 2 ; ≥ c3
⇒ ; ≥ ,c3 % c-3
,lower bound
⇒ ; ≥ ,c= % c>3
,lower bound
x % " 2 ; ≥ % c> % x 2 " 2 ; ≥ c= c5
; 0 c& C ;-3I 3 !-3 ⇒ x-I 0 x3I& C u 0 3 w ⇒ x I 0 a 0 -C. -3
-3
3
-
$f the networ# is not connected& then the problem decomposes into independent
problems& one for each component. component.
Exam*le 7 6our hospitals located within a cit" are cooperating to establish a centrali;ed blood%ban# facilit" that will ser!e the hospitals. The new fa facilit" cilit" is to be located such that the ,total distance tra!eled is minimi;ed. The hospitals are located at the following coordinates P-0,0,=&3& and P=0,-F&>. The number of deli!eries to be made per wee# between the blood%ban# facilit" and each each hospital is estimated to be >& J& 3& and -C& respecti!el". Kssuming rectilinear tra!el& determine the optimum location. m0= P- 0 , 0 ,=& 3
w- 0 >
P3 0 ,G& F
w3 0 J
w> 0 3
P= 0 ,-F& >
w= 0 -C
a> 0 =
w> 0 3
w> 0 3
a- 0 < a3 0 G
w- 0 > w3 0 J
w> 2 w- 0 < w> 2 w- 2 w3 0 -> ⇒
a= 0 -F
w= 0 -C
'omputation of xI
xI 0
'omputation of "I b> 0 3
w> 0 3
w> 0 3
b= 0 > b3 0 F
w= 0 -C w3 0 J
w> 2 w= 0 -3
⇒
"I 0
*0
b- 0 -C
w- 0 >
Exam*le 8 ,-C& -F
-F
6ind the optimal location of an ambulance with respect to four ,#nown possible
h= 0 --=
accident locations which coordinates are P-0,F&--& P30,-3&0,-=&G& and P=0,-C&-F. The ob9ecti!e is to minimi;e
-3 ,F& --
the maximum distance from the ambulance location to an accident location and from
-C
h- 0 -C
the accident location to its closest hospital. The distances from the accident locations to their closest hospitals are h-0-C& h30-F&
,-3& N J ,-=& G
h>0-=& and h=0--. Kssume that distances are rectilinear. rectilinear. $f multiple optima exist&
,-C& G
F
h> 0 -= ,-3& 0 -=
P3 0 ,-3& >rd set of points 0 4P3& P=& P
P=
Figure 0
P=′ P<
P3
P-
'3
P>
P-′ P3′
P=
Figure 4
P3
P′ =
P<
P-
'>
P>
P-′
P
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